Perfusion imaging provides the ability to measure variation in organ blood flow. Essentially, it comprises acquiring a sequence of images as a contrast agent flows in the organ of interest. This offers complementary information for diagnosing pathologies, and is particularly relevant in identifying liver lesions and malignancies. By way of a non-limiting example, computed tomography (CT) perfusion of the liver is herein considered. A hepatic perfusion study generally comprises at least three volume image acquisitions, corresponding respectively to the native (no contrast enhancement), hepatic arterial, and portal venous phases of the blood flow in the liver. See FIG. 1, in which (a) shows the native phase without contrast enhancement, (b) shows the arterial phase of enhancement, and (c) shows the venous phase of enhancement.
Such a complete study generally cannot practicably be done during a single breath-hold, and thus generally results in breathing artifacts. Cardiac and peristaltic motions can also cause geometric distortions between successive acquisitions. In order to compensate for the resulting soft tissue deformations, a suitable motion correction is required to obtain accurate perfusion measurements. See P. V. Pandharipande et al., “Perfusion imaging of the liver: current challenges and future goals,” Radiology 234(3), pp. 661-73, 2005. Such a motion correction generally comprises applying an image-based registration algorithm that finds an optimal non-rigid transformation aligning a pair of volumes by maximizing a suitable similarity measure. The specifics of liver perfusion imaging make this task particularly challenging. Typically, intensity values of some of the liver's tissue and arteries can be locally and drastically affected by the contrast agent intake. It is important to note that geometric distortions are highly non-rigid and can be of very large amplitude. In addition, the size and number of the volumes to be registered preferably requires a computationally efficient technique that performs the retrospective motion correction in an amount of time compatible with a clinician's workflow.
The use of standard registration techniques using the sum of squared differences to measure the quality of alignment tends to artificially shrink or expand contrast-enhanced structures. While one can partially mitigate this problem using more advanced statistical similarity measures, a different strategy is used herein to address this problem. Under certain simplifying assumptions, the liver can be modeled as an incompressible organ.
Non-rigid image registration techniques have been widely studied in the field of medical imaging. See, for example, D. Hill et al., “Medical image registration,” Physics in Medicine and Biology 26, pp. R1-R45, 2001; W. Crum et al., “Non-rigid image registration: Theory and practice,” British Journal of Radiology 77(2), pp. 140-153, 2004; J. Modersitzki, Numerical methods for image registration, Oxford University Press, 2004. A well-known intensity-based algorithm is the free-form deformation (FFD) described by Rueckert et al. “Non-rigid registration using free-form deformations: application to breast MR images,” IEEE Trans. Med. Imaging 18(8), pp. 712-721, 1999. The algorithm starts with an affine registration, to improve the capture range, and then recovers the non-rigid deformations based on a B-Spline model. G. E. Christensen et al., “Deformable templates using large deformation kinematics,” IEEE Transactions on Medical Image Processing 5(10), pp. 1435-1447, 1996, introduced an algorithm based on a viscous fluid partial differential equation (PDE), that captures large deformations.
Variations of the fluid method, to improve the computational costs, have been proposed in: M. Bro-Nielsen et al., “Fast fluid registration on medical images,” Visualization in Biomendical Computing 3, pp. 267-276, 1996; E. D'Agostino et al., “A viscous fluid model for multimodal non-rigid image registration using mutual information,” in MICCAI, pp. 541-548, 2002; and Crum et al., “Anisotropic multi-scale fluid registration; evaluation in magnetic resonance breast imaging,” Physics in Medicine and Biology 50, 5153-5174, November 2005.
As often used in computational anatomy (see M. I. Miller et al., “Geodesic shooting for computational anatomy,” J. of Mathematical Imaging and Vision 24(2), pp. 209-228), 2006, the registration process can be reformulated in a geometric context. See A. Trouvé, “Diffeomorphisms and pattern matching in image analysis,” Int. J. Computer Vision 28(3), pp. 213-221, 1998; and M. I. Miller and L. Younes, “Group actions, homeomorphisms, and matching: A general framework,” Int. J. Comput. Vision 41(1-2), pp. 61-84, 2001. Beg et al. reformulated Christensen's approach as a gradient flow on a group of diffeomorphisms. See Beg et al., “Computing large deformation metric mappings via geodesic flows of diffeomorphisms,” Int. J. Computer Vision 61, pp. 139-157, February 2005.
Material on the terminology of B-splines may be found in various published works such as, for example, in Medical Image Registration, by Joseph B. Hajnal et al., CRC Press, New York; 2001, pp. 65 and 286 et seq. and on affine concepts in the aforementioned book by Hajnal and, for example, in Geometry, by Dan Pedoe, Dover Publications, New York, 1970, pp. 351 et seq. and pp. 397 et seq.
Some of the previous techniques have been extended to handle various constraints. Registration is considered with topology preservation, see O. Musse et al., “Topology preserving deformable image matching using constrained hierarchical parametric models,” IEEE Trans. on Image Processing 10, pp. 1081-1093, July 2001; and V. Noblet et al., “3-D deformable image registration: a topology preservation scheme based on hierarchical deformation models and interval analysis optimization,” IEEE Trans. on Image Processing 14(5), pp. 553-566, 2005; or with local rigidity, see D. Loeckx et al, “Nonrigid registration using free-form deformations with a local rigidity constraint,” in MICCAI, pp. 639-646, 2004; and M. Staring et al., “Nonrigid registration using a rigidity constraint,” in SPIE Medical Imaging: Image Processing, J. M Reinhardt and J. P. W. Pluim, eds., Proceedings of SPIE 6144, pp. 614413-1-614413-10, SPIE Press, (San Diego, Calif., USA) February 2006.
T. Rohlfing et al., “Volume-preserving non-rigid registration of MR breast images using free-form deformation with an incompressibility constraint,” IEEE Trans. Med. Imaging 22(6), pp. 730-741, 2003 introduced an incompressibility constraint as a penalty term to the FFD model described. The constraint is defined as the integral of the absolute logarithm of the Jacobian determinant over the transformation domain.
In contrast, E. Haber and J. Modersitzki, “Numerical methods for volume preserving image registration,” Inverse Problems 20, (1621-1638), 2004 presented a volume-preserving constrained optimization, where they derived the Euler-Lagrange equations. Later, E. Haber and J. Modersitzki, in “A scale space method for volume preserving image registration,” in ScaleSpace, 2005 proposed a linearization of the constraint problem that solves a sequence of quadratic constrained optimizations. This amounts to the integration of the volume-preserving constraint in a fluid variational framework.
It is herein recognized that there exists a lack of efficient and reliable tools to cope efficiently with the foregoing issues.